Tuesday, June 23, 2009

The Egyptian ordinary cubit was divided into 24 fingers, and the Egyptian foot was 16 fingers, or 2/3 of a cubit. The cubit was the length from the tip of the finger to the elbow; thus it was essentially half a yard. Many ancient terms for this unit mean "elbow".

Surviving Egyptian rulers use the royal cubit which was an ordinary cubit plus four fingers, and this royal cubit was the basis of ancient Egyptian architecture and other measurements. A natural cubit is 6/7 of the royal cubit. The Greeks and Romans took up the natural cubit, while other ancient and Mesopotamian cultures continued the use of the royal cubit for sacred architecture.

Given the imprecision of ancient meauring tools, it's possible to only imprecisely compare ancient Egyptian measures to contemporary ones. The surviving royal cubit rulers are 526 mm, plus or minus 3 mm, so an ordinary cubit would be 450.8 mm. The customary American foot is 457 mm. If metric inch (25 mm) was used instead of the international inch (25.4 mm), the ancient Egyptian ordinary cubit would be exactly 18 metric inches.

This is a coincidence. The ancient Egyptian system of measurement was the basis of the Greek system of measurement, which was the basis of the Roman system of measurement, which was the basis for contemporary customary measurement. But the system shifted over time. The Romans emphasized the inch ("uncia") as a unit of measure, and the Roman inch and the Roman cubit were a few miliimeters shorter than the contemporary American standard. The word "cubit" itself comes from the Roman word for the unit, "cubitus."

Effectively, a cubit is a measure of 18 inches, a foot and a half, or half a yard, which in ancient times was closer to 450 mm.

Sunday, June 21, 2009

Many people know that the Maya calendar "predicted" a "catastrophe" in 2012. But the Maya calendar was strange and interesting. Rather than solar years, the Maya Long Count calendar counted the number of days from the beginning of their mythological epoch thousands of years earlier.

Counting the number of days in 4,000 years is somewhat unwieldy in our numeral systemit nearly a million and a half days. Rather than our decimal (base-10) system, the Mayans used a vigesimal (base-20) system of numbers. In vigesimal, each position in the numeral is worth more: rather than 1s, 10s, 100s, and 1000s, each position denotes 1s, 20s, 400s, and 8000s. Rather than count a million and a half days in the seven digits it would take in decimal numbers, they counted it in just five digits. The Long Count calendar was not a pure vigesimal count; the Mayans modified it so the first two digits would count a period of 360 days rather than 400 days, which was close enough to a solar year to be convenient for their purposes.

Imagine we used a system of counting days, and we only used five digits to do so. The day 13,765 would be the 250th day of the 37th Julian year. The day 99,999 would be the 285th day of the 273rd year. On the next day, the count would reset to zero, because we aren't using a sixth digit for the hundred thousandth day. In a decimal count, this would happen about every 273 years. In the Maya's modified vigesimal count, it would have happened every 2,880,039 days, or roughly every 7,885 years.

If, using our decimal system, we began counting the number of days since April 27, 1821, then December 20, 2012, would be the 69,999th day. The next day (December 21) would be the 70,000th day. Something analogous happens on that day in the Mayan calendar. Obviously, it's a great chance for someone to get rich selling stuff to New Agers.

A couple of months ago, I was working on a fictional puzzle that would use a vigesimal day-count calendar, which is an unfamiliar and confusing concept for most people, and was monkeying around with base-20 math and how to represent it graphically. One of the problems with thinking about numbers in other bases is that they're very difficult to conceptualize for a number of reasons. One is that we don't have many words to describe these numbers. Another is that we don't have adequate glyphs to represent them: there are only ten Arabic numerals.

Most of us have been trained for years in base-10 math, and find it very intuitive, easy to talk about, and easy to think in. The English language has remnants of alternative systems: base-12 or dozenal (dozens, grosses, and great grosses), and base-20 or vigesimal ("Four score and seven years ago..."). These systems are pretty common in the world's cutures, because they are easier to divide. While ten is evenly divisible only by two and five, twenty is divisible by 2, 4, 5, and 10, and twelve is divisible by 2, 3, 4, and 6. The base-60 or sexagesimal system of the Sumerians and Babylonians consisted of six base-10 units, and survives in measurements of time and geometry.

Computer scientists have been using octal and hexadecimal, base-8 and base-16, for decades because they are convenient ways to handle binary, base-2, numbers. But computer scientists don't have numerals to represent them, instead borrowing the letters of the alphabet to stand in for the missing digits. It can be a somewhat confusing system.

In 1968, Bruce A. Martin proposed a new series of numerals that drew out the 15 numerals according to their bit position. I tried some similar ideas, drawing out the numbers in a way somewhat similar to the Maya numerals. There is a design problem with this, though: the numerals are difficult to distinguish and sometimes rely on counting out every digit. In Martin's system, for example, the glyph for three is hard to quickly and clearly distinguish from the glyph for five. Good font design, or using an angle bracket instead of a straight line for the back, could help, but it's an inherent problem.

Working with invented alphabets and scripts, it's clear that a quick and easy way to generate glyphs for sounds is to flip or rotate existing characters. In the Latin alphabet, M, W, q, p, b, and d, are all flipped or rotated. So, I tried something similar for the Arabic numerals, and borrowed the character for 10 from kanji. It's certainly not a perfect system: 1 and 8 rotated look like other symbols, and need to be somewhat modified. Ten looks very like a plus symbol, and they would need to be clearly distinguished in a font. OTOH, for those of us accustomed to the decimal system, it's easy to remember which glyph represents seventeen.

Another reason that it's easy to think in decimals and hard in other bases is that the decimal positions all have names. Ten tens is a hundred, ten hundreds is a thousand, a thousand thousands is a million, a thousand million is a billion, a million million is a trillion. English has some words in dozenal: a dozen dozen is a gross, a dozen gross is a great gross. But what is a score score? I borrowed and twisted around some old Scottish and English words to have some vocabulary to play with.

A score scores is a dubbock; a score dubbocks is a skelling, and a score skellings is a skell. In decimal, a score is 20, a dubbock is 400, a skelling is 8,000, and a skell is 160,000. Thus, two skelling, twelve dubbock, nine score and sixteen is equivalent to (in decimal) twenty thousand, nine hundred, and ninety-six. Somewhere after a skell is a villion, the vigesimal million.

J.R.R. Tolkien's tengwar, used for writing his Elvish languages as well as English and other European languages, is truly a beautiful script: it is steeped in the calligraphic traditions of medieval European writing, elegantly and systematically represents most consonant sounds, has strong lines and a hypnotic, engaging repetition of basic shapes.

But, as a script, the tengwar causes me great frustration. Tolkien devised an extraordinarily elegant way of representing consonants, but he never really settled on a way of consistent or systematic way of representing vowels. There are full modes and tehta modes, all varying. Even the full modes barely have enough vowels for English, and the tehta modes are wholly inadequate. Furthermore, many fans of the scripts find phonemic English writing confusing, and so most people write in the so-called "orthographic" modes, which often write consonants phonemically and vowels according to their representation in the Latin alphabet.

Using the five vowels of the Latin alphabet to write the dozen-plus vowels of English is an ugly hack. Using the Latin vowels as written in English in an otherwise phonemic script is an ugly hack of an ugly hack.

Many people are intrigued by the idea of writing English in an abugida, and the tehta modes are probably as close as it's practical to come. But I find it pretty awkward. There are a two main problems with writing English this way: English has numerous consonant clusters, and English has too many vowels to use diacritics easily.

But it'd be great for Japanese! Japanese has a simple syllable structure, a simple vowel inventory, and is already written with a syllabary. So here is a Japanese mode for the tengwar.

These tengwar here generally follow rōmaji. The usage of súle for つ and anto for づ are unsystematic, but seemed like the best borrowing. The long vowel carrier is used in an unusual way; it is merely placed after a letter to show that the vowel is long. Note also the bottom bar used for palatized consonants, and I've offered no way to represent the sokuon っ.

このモードはロマ字みたい。本当は日本語を日本語の字で書くことがいいけど、テングゥアが面白い。

EDIT (July 26, 2009)Significantly revised, so one can write with a sokuon. Use the doubler bar (underbar) to mark a consonant that would be proceeded by a sokuon.

Thursday, June 18, 2009

The second is the unit of time we all know and love. It is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom at rest at absolute zero, corrected for gravitational time dilation.

A hectosecond is about a minute and a half.

A kilosecond is between 16 and 17 minutes. There are 3.6 kiloseconds in an hour.

A megasecond is about 11 and a half days.

A gigasecond is 31.7 years.

A terasecond is more than 31,000 years.

SI uses the second-minute-hour system because these standard prefixes are completely impractical. It's probably possible for people to think in a system that has 86.4 kiloseconds per day, and an 11-day week. But the system of metric prefixes doesn't allow for a unit of time between 11.5 days and 31.7 years. Some of the obsolete or nonstandard metric units might be helpful. A myriameter would be about 2.7 hours. A hectokilosecond would be 1.2 days.

Adapting an ordinary dial clock to indicate the time in kiloseconds would be a hassle, since the kilosecond doesn't divide easily into minutes or hours. But that doesn't mean we must ignore the humorous or unusual units of measurement. They can live on, as the moment does.

Impractical as it may be, I'm not ready to give up on the kilosecond. Fifteen minutes is a useful, common block of time, and a kilosecond is very similar. One one-hundreth of a day is 14.4 minutes, also very close. This is a unit of time that only needs a name: I dub it the brevel.

The brevel is a vague and indeterminate amount of time: not quite 15 minutes, but close; not quite a thousand seconds, but close; and there are nearly, but not quite, a hundred brevels in a day.

Tuesday, June 16, 2009

Cincinnati lies along the north bank of the Ohio River, between where the Great Miami and Little Miami Rivers flow into the Ohio. Dayton is about 55 miles north of Cincinnati on the Great Miami River, and Columbus is 110 miles northeast of Cincinnati, where the Olentangy flows into the Scioto River, a relict of the Teays. The Miami Rivers are so named because prior to American colonization, the rivers were the eastern boundary of the lands of the Miami (1). To the east and south (in Kentucky) were the traditional lands of the Shawnee around Chillicothe and the lower Scioto.

Morgan's Raid is the most significant engagements of the Civil War fought in Indiana and Ohio. The only remaining fortification remaining from the Civil War emergency defense of Cincinnati is Battery Hooper at the James A. Ramage Civil War Museum, which was filled in with dirt, in Fort Wright, Kentucky. The defense of Cincinnati was the occasion of the first bridge across the Ohio River there, a military pontoon bridge.

Monday, June 15, 2009

It's an entry-level commuter bike, but it's a smooth, nice ride. I have two other bikes: a full-size Dahon folder than is in terrible, wobbly shape from the many thousands of miles I have put on it over the last few years, and the Esquilax, an old steel 10-speed I pulled out of a dumpster and repaired to basic rideability, but have never gotten to good shape. At present, the rims are bent and warped, the wheels have numerous broken spokes, and the ball bearing retainer inside the bottom bracket is broken and grinding itself apart (still far better condition that I got it in!).

An Esquilax, of course, is the famous mythical creature that has the body of a rabbit...and the head of a rabbit. This new bike I have dubbed the Jub-Jub (a bird?), tho it is clearly un-fruminous.

Saturday, June 13, 2009

Jack Vance's Dying Earth series is populated by all manner of strange creatures, from the wee Twk-Men to the flying, carnivorous pelgranes. In chapter 5 of Eyes of the Overworld, while Cugel is travelling with the pilgrims, Cugel learns of Mad King Kutt of a past eon, who assembled a menagerie attested in the casebook of the wizard Follinense. From the little left of his notes, it is learned that...

Some of the modern chess variants like Berolina Chess aspire to be a serious direction for orthodox chess to go forward in the future. Aside from Fischer Random Chess, the most common variation of chess invented by skilled players for serious play includes two pieces, one that combines the moves of the bishop and knight and the other that combines the rook and knight, and a consequently larger board. These combination pieces date to at least 1617, but there is no consenus on whether to use one or both pieces, nomenclature, castling rules, initial setup, or board size. Capablanca Chess, developed by world chess champion José Raúl Capablanca (and played with Emmanuel Lasker), and Grand Chess are important contenders, although there are several others (1, 2, 3, 4, 5).

Chaturanga, of course, is the medieval Indian game that became contemporary chess after it passing through Persia as shatranj and into Europe. Dandelion shatranj is a minimalistic miniature shatranj inspired by mini-shogi (tho with somewhat less interesting gameplay) and named for the small, weedy flowers that pop up everywhere so easily. Gameplay and moves are exactly like shatranj, but occurs on a 5x5 board with only six pieces and a somewhat modified setup.

5 r e n c k4 - - - - p3 - - - - -2 P - - - -1 K C N E R a b c d e

King (K): Moves as in orthodox chess.Counselor (C): Moves one square diagonally in any direction. Knight (N): Moves as in orthodox chess.Elephant (E): Moves two squares diagonally in any direction, jumping over the intervening square. Rook (R): Moves as in orthodox chess. Pawn (P): Moves as in orthodox chess. When they arrive at the last rank on the board, pawns promote to counselors only.

Miniature shatranj has very limited play, but may be fun for a few games. There are only three or so viable opening moves, for example, and the elephant can only visit three squares on the board including its starting square. The king is fairly powerful, often threatening as many squares as the rook.

Mini-chaturanga is a group of variants on mini-shatranj, offering different early historical versions of the elephant's move. Standard mini-chaturanga is essentially the same as mini-shatranj, but the first player to expose a bare king wins without exception.

War Elephant Mini-Chaturanga: The elephant can move two squares orthogonally in any direction, jumping over the intervening square. This is the move of the Dabbabah, or war engine.

Silver Elephant Mini-Chaturanga: The elephant can move one square diagonally in any direction, and one square orthogonally forward. This is the move of the silver general of contemporary xiangqi and shogi, but in mini-chaturanga this piece does not promote.

Double Happiness Mini-Chaturanga: The elephant can move two squares orthogonally or two squares diagonally, jumping over the intervening square. This unhistorical move, as the alibaba, combines the move of the elephant and the dabbabah.

Wednesday, June 3, 2009

I think the first chess variant I was seriously interested in was Gary Gygax's dragonchess. It seemed like it'd be hard to get a set together, and gameplay'd be outlandish and complicated. And it is!

There are plenty of others, but I've played almost none of them. By far the most interesting, for a number of reasons, is Alice Chess. The pieces go through the looking glass every time they move. The most worthwhile is Fischer random chess or Chess960, invented in 1996 by Bobby Fischer.

There are a couple of four-handed chess games, aside from stuff like chaturanga and Enochian chess. I think the Dessau chessboard looks the most practical, as it eliminates pawn capture problems. Four-handed competitive play sounds more interesting than cooperative play. The Battle of Four Armies: capture an enemy king to force that player out of the game (their pieces cannot move or threaten, but are not removed from play until individually captured) and the last king standing wins.

The miniature shogi variants look like they give quick and elegant games. There are a couple of miniature versions of orthodox chess: HP Minichess, apparently presented by Marvin Gardner in Scientific American, is played on a 5x5 board. Toystore Chess is essentially miniature chess with placing. Quick Chess and Speed Game Chess are played on a 5x6 board.

The shogi variants are wild and impressive. Five-minute poppy shogi and mini-shogi are beautifully minimalistic, while taiyoku shogi is epically huge and unwieldy: there are 1296 squares on the board and each side has 402 pieces of 209 different types, totalling 253 different moves. Totally crazy.